midpoint

midpoint, a FreeFem++ code which uses the (implicit) midpoint method to approximate the solution of a time dependent boundary value problem (BVP).

Our problem is posed on the unit square, with zero boundary conditions, and having the exact solution

        u = sin(pi x) sin(pi y) e^(-t)
      

        du    d^2 u   d^2 u
        -- -  ----- - ----- = (2 pi^2 - 1 ) sin(pi x) sin(pi y) e^(-t)
        dt    dx^2    dy^2
      

        u(x,y,0) = sin(pi x) sin(pi y)
      

        u(x,y,t) = 0
      

Licensing:

The information on this web page is distributed under the MIT license.

Languages:

midpoint is available in a C version and a C++ version and a Fortran77 version and a Fortran90 version and a FreeFem++ version and a MATLAB version and an Octave version and a Python version and an R version.

Reference:

• Catalin Trenchea, John Burkardt,
  Refactorization of the midpoint rule,
  Applied Mathematics Letters,
  Volume 107, September 2020.
• Frederic Hecht,
  New development in FreeFem++,
  Journal of Numerical Mathematics,
  Volume 20, Number 3-4, 2012, pages 251-265.

Source Code:

• midpoint.edp, the FreeFem++ script;
• midpoint.sh, a BASH script to run the example;
• midpoint.txt, the output file;